In my bulk RNA-seq dataset, I noticed a few outliers on MDS plot and on further investigation found that these samples originated from one extraction batch (extbatch) 7. 

Fig 1: MDS plot showing outliers and that they belong to extbatch 7.

Most of the samples from batch 7 and even batch 8 had lower RNA yield than other samples. 

Fig 2: Mean and SD of RNA yield over extraction batches.

So now i am trying to correct this. I tried sva::ComBat() to correct for this extbatch and I ended up with negative values in my data. For comparison, I tried correction using scran::mnnCorrect. This also produced negative values.

Fig 3: Distribution of raw counts, counts after scran correction and combat correction. X-axis are individual samples. Y-axis are counts forced to limits -100K and 100K. Colours denote extraction batches. Red is the offending batch 7 with mostly bad samples. See appearance of negative values after batch correction.

https://i.imgur.com/wcIXtW3.jpg

Now, there is the argument that rather than correcting the batch, include it as a factor in my GLM model (~extbatch+family+condition). I did that and found 2 DEGs. If I remove samples from this batch 7 from my data altogether and run my model (~family+condition), I find many DEGs.

So, some of my questions are

1. Why do I get negative values and how do I fix that?
2. Is it better to discard these offending samples (I lose half my data) rather than fixing it using batch correction.
3. Why is the GLM with extbatch failing to produce any DEGs while removing the batch works?
4. If anyone has any better suggestions to handle this?

1. You get negative values because these methods, by and large, are designed to work on log-expression values that can go from -Inf to Inf. This is what they do, so there's nothing to "fix". If you want to account for the batch effect in a DE analysis, include batch as a factor in the design matrix.
2. According to your first MDS plot, samples in batch 7 are highly variable within the same batch. Batch correction won't help here, because your variability is occurring within the same batch. You need to figure out why this is the case before discarding those samples. For example, what are the DE genes between the batch 7 samples on the left and their counterparts on the right? Sex-related genes, or something?
3. See my answer to point 2. Including a batch effect won't solve your particular problem, because the variability is occurring within the batch. This will inflate the dispersions and reduce power. Removing the entire batch will avoid this effect - though obviously at the cost of losing samples from batch 7.
4. If you still can't figure out what's happening in batch 7, you could try using RUVSeq or sva to guess the factor of unwanted variation and put that in the model. However, there's no substitute for properly understanding what's happening with your data.